The Brezis–Nirenberg problem for the Laplacian with a singular drift in Rn and Sn

We consider the Brezis–Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both Rn and Sn, 3≤n≤5. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter (δ say) tim...

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Bibliographic Details
Published in:Nonlinear analysis 2017-07, Vol.157, p.189-211
Main Authors: Benguria, Rafael D., Benguria, Soledad
Format: Article
Language:English
Subjects:
Brezis–Nirenberg problem
Critical exponent
Drift
Drift Laplacian
Laplace equation
Mathematical analysis
Nonlinear equations
Rellich–Pohozaev identity
Spaces of constant curvature
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Summary:We consider the Brezis–Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both Rn and Sn, 3≤n≤5. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter (δ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space is characterized by the (geodesic) radius of the ball, δ, and λ, the coupling constant of the linear term of the Brezis–Nirenberg problem.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2017.03.006